Bifurcation analysis of an endogenous growth model

Abstract

Abstract:This paper analyzes the dynamics of a variant of Jones (2002) semi-endogenous growth model within the feasible parameter space. We derive the long run growth rate of the economy and do a detailed bifurcation analysis of the equilibrium. We show the existence of codimension-1 bifurcations (Hopf, Branch Point, Limit Point of Cycles, and Period Doubling) and codimension-2 (Bogdanov-Takens and Generalized Hopf) bifurcations within the feasible parameter range of the model.It is important to recognize that bifurcation boundaries do not necessarily separate stable from unstable solution domains. Bifurcation boundaries can separate one kind of unstable dynamics domain from another kind of unstable dynamics domain, or one kind of stable dynamics domain from another kind (called soft bifurcation), such as bifurcation from monotonic stability to damped periodic stability or from damped periodic to damped multiperiodic stability. There are not only an infinite number of kinds of unstable dynamics, some very close to stability in appearance, but also an infinite number of kinds of stable dynamics. Hence subjective prior views on whether the economy is or is not stable provide little guidance without mathematical analysis of model dynamics.When a bifurcation boundary crosses the parameter estimates' confidence region, robustness of dynamical inferences from policy simulations are compromised, when conducted, in the usual manner, only at the parameters' point estimates.

Natural cycles and pollution

“…The Bogdanov-Takens bifurcation (BT ) occurs when conditions for the saddle-node bifurcation and for the Hopf bifurcation meet each other. 2 At the Bogdanov-Takens point, the steady state becomes:…”

“…This unusual situation in econmics is known as a Bogdanov-Takens bifurcation. Recently, Barnett and Ghosh (2013) have pointed out that such a Bogdanov-Takens bifurcation is possible in an endogenous growth model. Our contribution adds a value to the existing literature by proving the existence of a Bogdanov-Takens bifurcation in an (environmental) exogenous growth model à la Ramsey.…”

“…The bifurcation is non-degenerate because the quadratic coe¢ cient associated with the normal form of the saddle-node bifurcation is a (0) = 0:04467276 6 = 0 (seeKuznetsov (1998), p.85, among others) 2. The BT bifurcation is non-degenerate because the two quadratic coe¢ cients associated with the normal form of the Bogdanov-Takens bifurcation are nonzero: (a (0) ; b (0)) = ( 0:1643409; 7:824758) 6 = (0; 0) (seeKuznetsov (1998), p320, among others).…”

Multiple equilibria and global indeterminacy in an endogenous growth model with congestible public goods

“…The novelty of our analysis is to detect the presence of global indeterminacy by making use of the Bogdanov‐Takens (henceforth, BT) bifurcation theorem, a powerful mathematical tool for simplifying highly non‐linear dynamic systems. The theorem is largely used in mathematics, physics and biology, but has found a surprisingly limited application in economics (see also Barnett & Ghosh, ; Bella & Mattana, ; Benhabib, Schmitt‐Grohé, & Uribe, ). Some examples are also provided to achieve concrete policy implications.…”

Bifurcation of Macroeconometric Models and Robustness of Dynamical Inferences

“…Eryilmaz [2013, 2014], investigated bifurcation in open economy models. Barnett and Ghosh [2013] investigated the existence of bifurcations in endogenous growth models.…”